Electrical computer



C. J. HIRSCH ELECTRICAL COMPUTER sept. 15, 1953 '7 Sheets-Sheet l lFiledMarch 2. 1948 FIG.2

2 :14| lill INVENTOR. CHARLES J. HIRSCH va y@ Time- ATTORNEY .FiledMarch 2, 1948 SePf- 15, 1953 c. J. HlRscH 2,652,194

v}5.LEcTRICAL COMPUTER 7 Sheets-Sheet 2 A f 2 o' d 3 FlRsT -I'- c 34 El'"ENERGIZING T|W.CONSTANT COMPARISON E, CIRCUIT o 0CIRGLHT RC d o CIRCUITo 2 --.f o @I I T|M|NG I TRIGGEREO 32 PULsE 4 PULSE I' GENERATOR a AGENERATOR ifA Si* '7) l). T1- 2 I I l -ENERGIZING SECOND I ETIME-CONSTANT SAMPL'NG i l-L I CIRCUIT r clROUITRc CIRCUIT I .g

TIME- y CONSTANT. clRcUlT R'c TINUNG PULSE J: GENERATOR Ef ENERGIzlNGCOLISMT'NT COMPARISON 2V E- o CIRCUIT v ^C|RCU|T R'C CIRCUIT 0 |5'TRIGGEREO 7 PULSE -III 85 GENERATOR ATTORNEY FIG..4

c. J. HlRscH ELECTRICAL COMPUTER Sept. 15, 1953 Filed March 2, `19415 7sheets-Cyan 3;

n0 nl n2 E E E oooto vooro uooso Time FIG.5

1m/nvm. CHARLES J. HIJRSCH BY l ATTORNEY Sept. l5, 1953 c. J. HIRscH2,652,194

ELECTRICAL COMPUTER Filed Maron 2, 194e 7 sheets-sheet `4 CHARLES HIRSCHFIG? vg ATTORNEY sept. 15, 1953 Filed March 2. 1948 C. J. HIRSCHELECTRICAL COMPUTER 7 Sheets-Sheet 5 Voltage Vohae IN V EN TOR.

CHARLES J. Hl RSCH ATTO R NEY Sept. 15, 1953 c. J. HlRscH ELECTRICALCOMPUTER 'r sheets-sheet e Filed March 2, 1948 ECC SOURCE -OF VOLTAGE ElCommmsoll CIRCUIT TRIGGERED PULSE GENERATOR SAMPLING CIRCUIT TIME-CONSTANT CIRCUIT RC' TIMING PULSE GEN ER ATOR Z E R O ADJUSTING CIRCUITENERGIZ I NG CIRCUIT SOURCE oF VOLTAGE SINGLE-SWEEP 97 SAW-TOOTH CIRCUITFIG.I2

INVENTOR. CHARLES J. HIRSCH BY JM/gi? ATTORNEY Sept. 15, 1953 c. J.HlRscl-l 2,552,194"

ELECTRICAL COMPUTER Filed March 2. 1948 7 Sheets-Sheet 7 CHARLES J.HIRSGH l ATTORNEY l f/ E /l Fnhn I I I. Ii T u. I m i V 3 I.. N f .I F0N c h D 2 WW .H E T )3 IU REA RC ESR AR GLE PI I GUN MC MPE .T G 4 i..\5 c I II... .II G i N n 01o U U W I P C 916 m m 2 L I I a.. E 7* s C L9J o c L M P I| Mf E EHT w. nw 2 wml E bmw Y m w .P GAWHC M 1| ALE* WS 4ILI. IIII S I.. ald, LILI Il G |.I-T I a I @I I U F H u H II II I I Emmm I R MNE N T E n O G E Patented Sept. 15, 1953 ELECTRICAL COMXUTERCharles J. Hirsch, Douglaston, N. Y;., assigner to Hazeltine Research,Inc., Chicago, Ill., a cor...

lwlation of lllinois Application March 2, 1948, serial No. 12,633`

(Cl.` 23S-51.)

20 Claims.

This invention relates to an electrical computer for solving equationsinvolving known and '1m-known parameters. A great many relation! shipsmay be expressed in the form of such equations, in which the knownparameters include one or moreindependent variables, some of which maybe assigned constant values in a particular case, and in which theunknown parameter is the dependent variable.

One general type of prior art computer, which `may be referred to as adigital computer, includes relay machinesy punch-card machines, andadding and multiplying machines utilizing either mechanical orelectronic counting devices. These computers can handle numerical dataafter theA problem has been reduced to a numerical routine susceptibleto solution by digital methods, which. often requires extensiveprogramming of the operation of the machine. The accuracy usually islimited onlyI by the number oi places to which a computation is carriedout, but the machine may have to perform a very extensive countingoperation to solve even a simple algebraic expression. Computers of thistype tend to be bulky and cumbersome in operation, particularly when theproblem is at all complex.

Another type of prior art computer may be classified generally as acontinuously variable computer. These computers deal with quantitiesbycontinuous correlation with mechanical dis- Tachometer inplacements orelectrical effects. struments come under this classification. Anotherexample of this type of computer is the resolver, in which a primarywinding carrying a voltage the amplitude of which represents a vector iscoupled to two secondary windings on a rotor mechanism. The rotor ismoved in such a waythat the coupling of the primary winding to these twosecondaryT windings varies as the sine and cosine respectively of theangular direction of the vector. Thus the amplitudes of the voltagesinduced in the two secondary Windings may represent respectively thecomponents of the vector as projected on the axes of a system ofCartesian co-ordinates.

Compared with digital computers, the continuously variable computersusually have the advantage of high speed and facility of setting up thecompu-ter to solve a given problem, but have the disadvantage that theiraccuracy tends to be lower. In order to provide a computer of thecontinuously variable type to solve a particular problem, it isnecessary to nnd an effect which can be made to follow the independentvariables involved in the problem continuously with, proper trackingandY without objectionable backlash or time lageects. Much ingenuityhas` been exercised to devise mechanical, electrical, orelectromechanical devices suitable for accomplishing these purposes andfor providing a useful indication of the result of the computation. Ingeneral',` however, each such computer can be used to solve only a veryrestricted form of problem, and hence usually is permanently coupledmechanically or electrically t0 the Source of the independent variableinvolved in the computation. This specialization of function, dictatedby the special nature of the mechanical or electrical devices utilizedin the computer, makes the continuously variable computers o i limitedusefulness in the solution of the mathematical problems or algebraicexpressions most frequently encountered.

Accordingly, it is anV object of the present invention to providel a newand improved electrical computer which substantiallyA avoids one or more0f the limitations and disadvantages of prior arrangements of the typedescribed.

It is also an object of the invention to provide a new and improvedelectrical computer applicable generally for solving equations` thesolutions of which involve the common mathematical relationships.

It is. al further object of the invention to provide. a new and,improved electrical computer free of mechanical moving parts,v compactand of small weight, yetcapable of computations at high speeds.

It is a still further object of the invention to provide a new andimproved electrical computer capable of continuously and rapidlyrecalculatins a problem involving parameters subject to changes.

It is Still another object of the invention to provide a new andimproved electrical computer for performing algebraic operations inwhich all of the independent and dependent variables are represented byvoltages referred to a convenient reference or datum voltage.

`In accordance with the. invention, an electrical computer for solvingequations involving known and unknown parameters comprises an electricalreference circuit having circuit elements with resistance and reactancevalues so proportioned as to develop an electrical eiect varying as apredetermined time function the value of which at some time represents aknown parameter of an equation to be solved; a source of potentialadjustable to a value representative of a parameter of said equation;and means for utilizing vthe adjustable potential and the effect at thethe predetermined time to develop a control effect. The computerincludes an electrical controlled circuit having circuit elements withresistance and reactance values so proportioned as to develop anotherelectrical effect having a value electrically independent of the valueof the first-mentioned effect and varying as a predetermined tirnefunction a value of which at some time is related to the value of anunknown parameter of the equation. The computer also includes meansresponsive to the control effect for evaluating the other effect toderive a resultant effect which represents an unknown parameter of theequation, and means for utilizing the resultant effect representative ofthe unknown parameter.

For a better understanding of the present invention, together with otherand further objects thereof, reference is had to the followingdescription taken in connection with the accompanying drawings, and itsscope will be pointed out in the appended claims.

in the drawings, Fig. 1 is a circuit diagram of an electrical computerembodying the present invention; Fig. 2 is a graph utilized inexplaining the operation of the Fig. 1 arrangement; Fig. 3 is a circuitdiagram, partly schematic, of a modified form of the Fig. 1 arrangementwhich has a Inode of operation also represented by the graphs of Fig. 2;Figs. 4, 6, 8, l0, 12, and 14 are circuit diagrams, partly schematic, ofother embodiments of the invention; while Figs. 5, 7, 9, l1, 13, and 15are graphs utilized in explaining the operation of the last-mentionedmodified forms of computer.

Referring to Fig. 1 of the drawings, there is shown a circuit diagram ofan electrical computer for solving equations involving known and unknownparameters and particularly well adapted for performing the mathematicaloperation of raising a number to a power either greater or less thanunity. The computer comprises a plurality of circuits in the form ofenergy-storage networks each effective upon energization to produce ameasurable effect having a value which is a predetermined function oftime, the known and unknown parameters being represented by valuesassumed by the time functions of these effects at related times. One ofthese networks is a reference circuit I I comprising a shunt combinationof an adjustable resistor R and an adjustable condenser C. This networkis effective to develop a voltage effect varying as a predeterminedfunction of time, specifically an exponential decay-time function, thevalue of which at some time represents a known parameter. The voltageeffect Vdeveloped across the network II has a value which is solelydependent upon the value of the initial voltage applied to the networkand the predetermined timeconstant characteristic thereof. Theenergication of the network II is controlled by a source of voltage E1',in the form of an adjustable battery I2 which is coupled to an outercontrol electrode of an energizing-circuit vacuum tube I3 the anode ofwhich is connected to a source of space current indicated as +B. Thecathode circuit of the tube I3 includes the network II.

The computer also includes another energystorage network I comprising anadjustable resistor R" and shunt-connected adjustable condenser C. Thisnetwork is effective upon energization thereof, at a time related to thetime of the energization of the reference circuit II,

to develop another voltage effect having a Value electricallyindependent of the value of the firstmentioned effect and varying as apredetermined exponentially decaying time function. In a manner similarto that mentioned above in connection with the network II, the voltageeffect developed across the network I5 has a value which is solelydependent upon the value of the initial voltage applied to thelast-mentioned network and the predetermined time-constantcharacteristic thereof. Therefore, the value of the effect developed ineither network II or I5 is electrically independent of the value of theeffect developed in the other network. The energization of the networkI5 also is controlled by a source of a voltage E1, in the form of anadjustable battery I5, coupled to another energizing-circuit vacuum tubeI1. This vacuum tube has an outer control electrode to which the batteryI5 is coupled and an anode which is energized from a space-currentsource +B. The cathode circuit of the tube Il includes the network I 5.

A timing-pulse generator I8 is provided for repeatedly energizing boththe reference network II and the controlled network I5. The unit I3 is amultivibrator having two vacuum tubes I9 and 2li coupled to a source ofspace curtrodes of the two energizing tubes I3 and II.

The computer arrangement also includes means responsive to the value ofthe effect developed in the reference network II at a particular timeafter each of the repeated energizations thereof for evaluating theother effect developed in the Y network I5 to derive an effect whichrepresents an unknown parameter. This means comprises a comparisoncircuit 5I coupled to the network il and a triggered pulse generator 32having an input circuit coupled to the comparison circuit 3l and havingan output circuit coupled to an input circuit of a sampling circuit 33.The sampling circuit also has an input circuit coupled to the controllednetwork I5.

The comparison circuit 3| is provided with a source of voltage E2 in theform 0f an adjustable battery 34. The comparison circuit 3i alsoincludes a tube 35 having a cathode-load resistor 36. The controlelectrode of the tube is connected to the battery 34, while the anode isconnected to a source of space current +B. The cathode of the tube 35also is coupled through a resistor 3`I to the anode of a diode vacuumtube 33 the cathode of which is coupled to the network II. The anode ofthe tube 38 is coupled through a condenser 39 to the anode of a vacuumtube III in the pulse generator 32.

The pulse generator 32 is of the blocking oscillator type. The anode ofthe tube 4I is coupled through one winding of a transformer 42 to asource of space current +B. Another winding of the transformer i2 hasone terminal connected to the control electrode of the tube 4I and theother terminal coupled through a resistor 53 to a source of biasingpotential indicated as -C. The resistor 43 also is connected to apulse-forming delay line 44 the remote terminals of which areopen-circuited. If the output signal from the comparison circuit 3l hasinsufcient magnitude, a pulse. amplifier may be inserted between thecondenser 33 and the anode oi tube 4 I.

The sampling circuit 33 is of the bridge-rectiier type and comprisesfour diode vacuum tubes 'I, 48, 49, 56 arranged in a conventionalbridgerectier circuit. One pair of diagonal terminals of this bridgecircuit is connected between the network I5 and a grounded resistor 52.The other pair of diagonal terminals of the bridge is connected to anactuating circuit comprising an output winding 45 provided on thetransformer 42 of the unit 32. The actuating circuit also includes inseries with the winding 45 a source of bias potential 53.

For utilizing the effect representative of the unknown parameter, thiseiiect being a voltage Ez" derived across the resistor 52, there also isprovided a vacuum-tube voltmeter 54. rlhis volt meter includes a vacuumtube 55 having a control electrode coupled to the resistor 52 and ananode connected to a source of space current +B. The cathode circuit ofthe tube 55 comprises a condenser 55 across which is connected avoltmeter V for reading the voltage E2". rThe capacitance of condenser55 and the resistance of the voltmeter V have values so selected as toprovide for the latter element a suitably large time constant.

The operation of the computer just described will now be described withreference to the curves of Fig. 2. The timing-pulse generator i8operates in a conventional manner to generate across the resistor 35 aseries of timing pulses of negative polarity represented by curve A. ofIn the computer described, lthese pulses repeat at a regular rate, andthe time of starting of one such negative pulse is identied by the timet1. These negative pulses are applied to the inner control electrodes ofthe energizing tubes I3 and Il. The pulses have a magnitude ils sulncient to cause these tubes to become remain nonc-onductive for theduration of the negative timing pulses regardless of the voltages whichmay be applied to the outer control electrodes of tubes I3 and Il andeven when the voltages developed across the networks II and I5 are quitesmall. Prior to the start of each negative timing pulse, the innercontrol elec trodes of the tubes I 3 and I1 have a positive pulseapplied thereto from the generator I8 and thus permit the now of spacecurrent through the respective networks I I and I5. In the case of thenetwork ll, the condenser C is charged up to and maintained at thevoltage E1 of `the battery I2 by virtue of the well-knowncathode-follower mode oi operation of the tube i5. vAt the time t1 whenthe tube I3 is effectively disconnected from the network II by beingbiased to anode-current cutoff as above explained, the condenser C'commences to discharge exponentially through the resistor R in themanner represented by curve B of Fig. 2.

Meanwhile, the tube 35 also operating as a cathode follower developsacross its cathode-load resistor 36 a voltage E2 equal to that appliedto the control electrode of the tube 35 from the bat tery 34. Thus, thedifference voltage between the voltage of the network I I and thevoltage developed across the cathode resistor 36 appears across theseries combination of the resistor 3l and the diode 38. When the Voltageacross the network II drops just below the voltage Ez appeering acrossthe resistor 36, the diode 38 conducts to develop across the resistor 31a pulse of voltage having negative polarity. This latter voltage isapplied through the condenser 39 to the pulse generator 32 at a time t2,as represented by curve C` of Fig. 2. rlhe leading edge of this pulse atthe time t2 initiates the generation oi a single potential pulse ofshort duration in the pulse generator 32. The duration of the pulse thusgenerated in unit 32 is determined by the time required for a negativepulse to be reflected from the open end of the delay line 44 back to thecontrol electrode of the tube 4I, and rthis duration is chosen justgreat enough to provide Y for adequate sampling of the voltage presentacross the network I5 at and immediately after the time t2,

The resulting pulse from the unit 312, represented by curve D of Fig. 2,is applied from the output winding 45 of the transformer 42 to thesampling circuit 33 with proper polarity and magnitude to overcome thebiasing voltage E@ of the source of potential 53. Accordingly, thebridge circuit 4l, 48, 49, 50 is actuated and rendered conductive forthe duration of the applied pulse with the result that the voltage thenexisting across the controlled network I5 appears across the resistor52. The controlled network I5 is energized under control of the tube I1in essentially the same manner as the network II is energized undercontrol of the tube I3. Also, the network I5 is energized at the sametime t1 as is the reference network I I. Its Voltage at the time ofenergization is the voltage E1" of the adjustable bate tery I3. Thevoltage wave form developed across the network I5 is represented by thecurve F of Fig. 2. After energization at the time tl, the exN ponentialdecay continues until, at the time t2, the voltage across the network I5has reached a value E2. It is at this time that the sampling circuit 33is actuated, in a manner previously dem scribed. in response to thevolta-ge E2' appearing in the network II so that the voltage developedacross the resistor 52 of the sampling circuit also has the value E2.The voltmeter circuit 54 operates as a cathode follower and itscondenser 5S is charged to the voltage E2. The resulting voltage E2 asmeasured by the voltmeter V is represented by the curve G of Fig. 2.

It is to be understood that the value E2 which the voltage across thenetwork I 5 reaches at time t2 is dependent solely on the value E1" ofthe voltage initially applied to this network and the predeterminedtime-constant characteristic of the network I5. The values E2 and E2 areobtained electrically independently of one another,

the circuit or network developing the one in no way affecting thecircuit or network developing the other.

The manner in which the Fig. l arrangement is utilized to performcomputations will more fully apparent from the following mathematicalanalysis of its operation.

The voltage across the network II decays from the value E1 at time t1 tothe value Es at time t2 in accordance with the expression:

where R'=the value of resistance of the resistor R C=the value ofcapacitance of the condenser C Likewise, the voltage across the networkI5 decays during the same interval from the value The time constants ofthe networks I and il are related by the expression: i

IIIC1II=I7IZCII It can be shown that the simultaneous solution ofEquations l, 2 and 3 is given by the relation:

E l/ E n El.; 4)

Thus, to solve an equation of the form:

it is necessary only to adjust the batteries I2 and I6 so that thevoltages E1 and E1" are equal to unity on any suitable voltage scale.The Equation 4 then reduces to the form:

The voltage E2 thereupon represents the parameter y and the voltage E2represents the parameter :c of Equation 5. As an example, let .r inEquation 5 equal 0.6, in terms of E1=E`1= and n equal 2.1. The timeconstant of the network I I is given a value relative to the timeconstant of the network I5 by suitable adjustments of the values of theresistors R', R and the condensers C', C" such that the ratio n of thetwo time constants equals 2.1, to satisfy Equation 3 above. voltage E2equal 0.6 on the same scale on which the voltages of the batteries I2and i5 were set to unity. The voltage E2, as read on the meter V, thenis found to be 0.34 which is the value of the dependent-variableparameter y of Equation 5.

Another method of using the arangement of Fig. 1, when :1: in Equation 5is greater than unity, lies in making a suitable choice for the value ciE1" in Equation 4. This equation can be rewrit ten as follows:

E2 El becomes:

Erf: om 8 Thus, assume that n: in Equation 5 has a value of 5 and n avalue of 2.1. Further, let El1=10 volts so that E1"=1021=126 volts; thenthe voltage E2 as read on the meter V is found to be 29.5 volts which isthe value of the dependent-variable parameter y of Equation 5. Theoperation, expressed in a different manner, may be said to be that thevoltage E1" having the initial value of' 126 volts decays to the valueyzE'zfzZQ volts in the same time that the voltage E1' of initial valueequal to volts decays to =Ez=5 volts.

Proper attention to several details of design, readily apparent to thoseskilled in the art, suffices to give an indication of the desiredsolution of a mathematical equation within. the limitsofaccuracyordinarily desired. For example, the scales on which the voltages of thebatteries I2, I6, and 34 are read may be compensated to take intoaccount the fact that the voltages across Battery 34 then is adjusted tomake the.

if then E1" is made equal to (E1' Equation 7- the resistors R', R, and36 at time t1 may be slightly different than the voltages E1', E1", and'E2', respectively. A further compensation of the voltage scale used inadjusting the battery 34 to the proper voltage may be desirable becauseof the slight difference in voltages between the voltage of the networkII and the voltage across the resistor 35 required to cause the diode 38to conduct.

voltage drops appearing across the rectiers 4l,

48, 49 and 50 of the sampling circuit 33 and the slight discrepancy dueto the discharging of con-` denser 56 between cycles of computeroperation. Also, in designing the sampling circuit 33 it may bedesirable to neutralize carefully the reactances in the several arms ofthe bridge. of computation is enhanced by so choosing the time constantsof both of the networks I I and I5 that the time t2 at which thesolution is derivedI occurs before the networks have become dis= chargedto the ilatter portion of their discharge' curves. Similarly, theaccuracy of computation is improved with increasingly larger values ofthe several reference voltages, thus minimizing stray voltage drops inthe computer circuits.

In the rst example given above of the solution of an equation of theform of Equation 5 the values of :c and hence of y are less than unity,and this would be the case even though the power n were less than unity.When the value of :n ands hence of y in Equation 5 is greater thanunity, regardless of the value of n, it is sometimes more convenient touse a circuit arrangement of the type shown in the partly schematiccircuit diagram of Fig. 3.

Most of the circuit arrangement of Fig. 3 is the same as that of Fig. 1,elements which are the' rameter so that the voltage E1 to which thenetwork I5 is charged initially must be adjusted to the value necessaryto obtain the preset value of E2. Accordingly, there is included asource I6 of voltage controllable to provide the requisite voltage E1.

The unit I6 comprises a battery 62 having a voltage E and connected toan input circuit of the energizing circuit I'I through a voltagedroppingresistor 63. The resulting voltage E1" at the input circuit of theenergizing circuit Il is read on a voltmeter 54.

There also is provided a source of voltage E2, 1n the form of anadjustable battery 64, and a control circuit 66 is included foradjusting the voltage E1 provided by the unit I6. In the control unit66, the voltage appearing across the output circuit of the samplingcircuit 33 is developed across a tapped resistor 6'! in series withV aresistor 68. The resistors 6l and 38 together have such a highresistance that the rate of diS- charge of a condenser 69 connectedacross the two resistors in series is low. The unit 66 also includes twovacuum tubes 'II and l2, the cathodes of which are connected to thejunction of resistors 61 and 68. is connected to the output circuit ofthe source I 6 while its control electrode is connected to the tap onthe resistor 61.

Likewise, the scale of the voltmeter V mayy be compensated to take intoaccount the small The accuracy The anode of the tube 'II` The anode 0fthev tube 12 is connected to a source of space cur' 9 rent +B, While itscontrol electrode is connected to the battery E4.

When the computer illustrated in Fig. l3 is turned on, the voltage E2 ofbattery 64 appears across the cathode load 68 of the tube 12. Assumingthat no space current is being drawn aS yet by the tube 1|, there is novoltage drop in the dropping resistor 63 and a high voltage E is appliedto the energizing circuit |1. This results in a voltage derived in thesampling circuit and obtained across the condenser EG which is greaterthan the voltage E2 across the resisto? B8. This causes current to flowthrough the resistor 61, thus biasing the control electrode of the tube1| to a positive Voltage and permitting that tube to conduct and passcurrent through the resistors 63 and E8. The resulting voltage drop inthe dropping resistor 63 thus increases during the first few cycles ofcomputer operation until the voltage E1" becomes low enough that thevoltage derived in the sampling circuit 33 practically equals thevoltage E2", as desired.

As an example of the operation of the Fig. 3 arrangement to solve anequation of the form of Equation 5, reference again may be had to `thecurves of Fig. 2. It is necessary only to adjust the batteries 3d and $5to that the voltages E2', curve B, and E2", curve F, are equal to unity.Equation 4 then reduces to the form:

The voltage Ei thereupon represents the parameter y and the voltage Eirepresents the parameter .r oi Equation 5. For example, let r equal 1.6and n equal 2.1. As before, the time constants of the networks and l5are adjusted to make the ratio n of the two time constants equal 2.1.Battery i2 then is adjusted to make the voltage E1 equal 1.6. Thevoltage E1, as read on the voltmeter 54', then is found to be 2.7 andindicates the value of the dependent-variable parameter y of Equation 5.Curve G of Fig. 2 in this case represents the voltage developed acrossthe condenser 6.9 of the control circuit 66.

By making suitable choice of the voltages Ei', E2', E1, E2", and settingthe value 1t in Equation 4 equal to unity, it will be `apparent thatequations of various forms may be solved. For example, if the ratio n ismade unity and the voltage E1 of the Fig. 1 arrangement is adjusted tobe unity, Equation 4 reduces to the form:

which accordingly may represent the parameters of the equation:

When both of the parameters v:r and y are Agreater than unity, however,it is desirable to set E2 rather than E1 equal to unity and in this casethe circuit arrangement of Fig. `3 should be used. Likewise, by makingE2 equal to unity Equation 4 may be rearranged to the form in which itrepresents the parameters of an equation having the form:

.'y Here again the Fig. `3 arrangement may be used if y is less thanunity, or Equation 4 may be changed to the form:

J2Il ZLWzI/(Ell/ljlll) which is of the Equation 12 form wherein Fig. 4is a partly schematic circuit diagram illustrati-ng a modiiication ofthe Fig. 1 arrangement particularly suitable for the solution ofequations of the form of Equation 10. A battery i2 is a source of aninitial voltage preferably higher than any of the voltages representingparameters involved in the computation. The battery |2 is coupled to theanode of a triode vacuum tube i3', which is included an energizingcircuit for the unit Ii and has its control electrode coupled to theoutput circuit of the timing-pulse generator I 8. Referring to curves of5, illustrative of the operation of the Fig. i arrangement, a timingpulse of negative polarity delivered by the unit iii` is represented bycurve lei `starting at the time to. Prior to this time the `@Ondenser C"of the time-constant circuit R, C of a circuit has been charged to anysuitable initial voltage E0". The circuit I i then commences at time toto discharge exponentially, as represented by curve J of Fig'. 5.

The circuit is coupled to a comparison circuit 3|', which is similar tothe comparison circuit 3| of Fig. 1 except that the resistor 31 anddiode 3B' of the circuit 3| occupy the places of the diode 38 andresistor 31, respectively, 4of the circuit 3| of Fig. 1. When thevoltage of the time-constant circuit falls just below the voltage E1" ofthe battery 34, a pulse of positive polarity is developed across theresistor 31 and is applied through a series condenser 39 and a shuntresistor 33 to the input circuit of a univibrator 15. The wave form ofthe pulse so applied is `represented by curve K of Fig. 5. The time ofcommencement of this pulse is identified aS t1.

The univibrator 15 is of conventional design and includes two vacuumYtubes 16, 11 having a common cathode resistor 18. The anode of the tube16 is coupled to a source of space current |B through a load resistor 19and is coupled to the control electrode of the tube 11 through acondenser 8|. The latter control electrode is so biased as normally to.cause space-current conduction in the tube 11. To this end, the controlelectrode of tube 11 is coupled to a source of biasing potential +Bthrough a resistor 32. The anode of the tube 11 is connected directly tothe space-current source +B. When a positive pulse is applied by theunit 3| to the input circuit of the univibrator 15, the vacuum tube 16is made conductive, and a pulse of ne-gative polarity represented bycurve L is derived at the anode of the tube 16 at the time i1 andapplied through a condenser 84 to a resistor 35 coupled across an inputcircuit of the energizing circuit I1. Another input circuit of theenergizing circuit i1 is connected to a source of a voltage E1 in theform of an adjustable battery I6. The energizing circuit l1 serves toAenergize a time-constant circuit R', C in a unit l5 which at time t1initiates an exponential discharge, represented by curve M of Fig. 5,startn ing at the voltage E1.

When the voltage of the circuit i5 falls to a value E2', as determinedby the voltage E2' of an adjustable battery 64 controlling a comparisoncircuit 3|, the comparison circuit derives a pulse of `negative polarityrepresented by curve N. This pulse triggers the pulse generator 32 toprovide a pulse of short duration having the wave form represented bycurve P. The latter iilse is applied to actuate a sampling circuitMeanwhile the timing-pulse generator i8' has applied, at time to, atiming pulse of negative polarity to an energizing circuit 86 coupled tothe battery I6'. By this means the voltage of an additionaltime-constant circuit R, C'" of a unit 91 is brought to a value Eo"prior to the time tt, at which time an exponential decay of the voltageacross the circuit 81 commences as represented by curve Q. The samplingcircuit 33 is coupled to the circuit 81 and at time t2 is actuated, asdescribed hereinabove, to derive in its output circuit the voltage Ez"represented by curve R of Fig. 5. This voltage is indicated on thevoltmeter 54.

The manner in which the arrangement of Fig. 4 is utilized to performcomputations Will be more fully apparent from the following mathematicalanalysis of its operation.

Referring to curves J, M, and Q of Fig. 5, it will be apparent that:

Equation 16 may be evaluated by substituting in it the value of t2obtained from Equations 14 and 15. From Equation 14:

E IH=E lll 2 0 EDI/Ell which simplifies to Equation 4 if (E1/Eo") is Fmade equal to 1, if E2'" is made equal to E2", and if Eo'" is made equalto E1". Letting n=1, Equation 25 becomes:

l2 and if Eo" is madeequal to E0 El, then E2III=E1IIE2I- Which is a formof Equation 11 Without any restrictions. Thus to multiply 5 by 8, letE1=5, E2=8, EO"=E1=10, Eu'=100, and

5X8 lll E2 --l00[ 10X10 40 (28) In other Words, Eo" will decay from 100to 40 volts in the total time required for Eo" to decay from 10 to 5volts plus the time required for E1 to decay from l0 to 8 volts.

The block diagram of Fig. 6 represents another modied form of theinvention essentially similar to that of Fig. 1. In the Fig. 6arrangement, a source of a conveniently high initial voltage En providedby a battery IZ' is applied to an energizing circuit I3' controlled froma timingpulse generator I8 so as to energize a timeconstant circuit R',C in a unit Il'. Simultaneously the timing pulse from generator I8' isapplied to a triggered pulse generator 32 to develop a pulse of shortduration which is applied to an adjustable delay circuit 9i This delaycircuit includes a rod or strip 92 of magnetostrictive material havingenergy-absorbing clamps 93, 93 at the ends thereof. The output circuitof the pulse generator 32 is coupled to a coil 94 magnetically coupledto the strip 92 and adjustable therealong. An output coil 95, alsomagnetically coupled to the strip 92, is coupled to the input circuit ofa pulse-shaping circuit 97 of conventional construction. The inputcircuit of the pulse-shaping circuit 91 may include a source of biasingcurrent for the output coil as is desirable to obtain a desirablemagnetic condition in the magnetostrictive strip 92 in the region of theoutput coil 95. Time-delay devices having a construction suitable forthe unit 9| are described and claimed in the following copendingapplications: Alan Hazeltine, Serial No. 785,248, led November 12, 1947,entitled Magnetostrictve Signal-Translating Arrangement, now Patent No.2,526,229; Theodore J. Fister, Serial No. 785,313, led November 12,1947, entitled Magnetostrictive Converter, now abandoned; and Leslie F.Curtis, Serial No. 785,425, filed November 12, 1947, entitledMagnetostrictive Time-Delay Device, now Patent No. 2,455,740, allassigned to the same assignee as the present invention.

Referring to the curves of Fig. 7, the timing pulse of negative polaritydeveloped in the generator I8 of Fig. 6 is represented by curve S. Theleading edge of the negative timing pulse causes the generation at atime to of a pulse of short duration in the pulse generator 32', asrepresented by curve T. This pulse is applied to the delay circuit 9|and emerges from said circuit at a time t2 determined by the positioningof the coil 94 along the magnetostrictive strip 92. After Wave-formcorrection in the pulse-shaping circuit 91, the delayed pulse induced inthe coil 95 has the form represented by curve U and oc'- curs at thetime t2.

Meanwhile the voltage of the circuit Il', represented by curve V, hasbeen decaying from the value Eo at the time of energization to to reachat a time t1 the Value E1 determined by the adjustment of an adjustablebattery 34. The circuit II and the battery 34 are coupled to acomparison circuit 3l', so that the latter develops a control pulse atthe time t1. This control pulse is represented by curve W ofL Fig. 7 andcauses the generation of a pulse, represented by curve '13 X, in aunivibrator to which the comparison circuit 3| is coupled.

The leading edge of the pulse generated in the univibrator 15 energizesat the time t1 the timeoonstant circuit R", C in the unit I5, throughthe action of an energizing circuit I1 having an input circuit coupledto the univibrator 15 and an output circuit coupled to the circuit I5. Avoltage Ei provided by an adjustable battery I6 is applied to anotherinput circuit of the energizing circuit I1, so that the voltage in thecircuit I5 at the time of its energization is E1. At the time t2 thedelayed pulse, curve U, of the pulse-shaping circuit 91 actuates asampling circuit 33 coupled to the time-constant circuit I5 to derivetherefrom a voltage E2, which is indicated on the voltmeter 54. Thecurve Y represents the voltage of the circuit I5, and the curve Zrepresents the voltage E2 derived in the sampling circuit 33.

At the time t2 the voltage of the circuit I has decayed to a value E2',assuming no disturbance in the circuit II due to the action of thecomparison circuit 3| at the time ti. With this assumption the voltageof the circuit has the values E1 and E2 at the times ti and t2,respectively, while the voltage of the circuit 5 has the values Ei" andE2" at the corresponding times. Therefore the relationship of Equation 4applies to these four voltages. Assuming the time constants of the twocircuits and I5 to be the same., the Equation 4 reduces to the form:

If the voltages Ei" and Ez both remain unchanged during a series ofcomputations, Equation 29 may be given the form:

MDE?? (am ch a case Ei may be preset by a suitable rdjilstment of thebattery It. In using the circuit arrangement of Fig. 6, however, it isnot necessary to inject the voltage E2 into the circuit at all. Instead,the time interval (tzto) is predetermined, either by calculation usingan equation similar to Equation 1 with predetermined values of Eo' andE2 or b y measurement of a circuit the saine as the circuit II' butwithout the possibly disturbing influence .of a coinparison circuitcoupled thereto. If this time interval is determined by calculation, thevoltage E2' may happen to occur at a point of the exponential dischargeof the condenser C where the slope of curve V is relatively small sothaJ the circuit would be insensitive to physical measurements at thetime t2. En is a voltage greater than any value which the voltage E1 mayassume and, although arbitrary, must be maintained constant duringsuccessive computations because it alects the time interval t2-to. Acontrolling pulse representing the internal tz-tu is obtained in theFig. 6 arrangement by the setting of the movable coil 9400i thedelayhcircuit 9| which determines the time t2 at which the pulse ofcurve U occurs. In this way the sampling circuit 33 of the Fig. 6arrangement is made effectively responsive to a value E2' of the'voltagein the circuit Il' at 1time tz, although this value or contro purposes.1sZrtotitlsiesddesired to solve with the present arrangement an equationof the form:

the voltages E2 and Ei" of Equation 30 may be used as representative ofthe parameters l1,/ Vand respectively, of Equation 31. If the values ofEi" and Ez', Equations 29 and 30, are chosen so that the constant lcequals unity, then the Fig. 6 .arrangement maybe used to determine thereciprocal y of the parameter a: of Equation 3l.

Fig. 8 is a circuit diagram, partly schematic, representing an.embodiment of the invention useful for Aperforming addition orsubtraction. In this arrangement, all of the variables involved are`represented by a voltage relative to ground which is proportional tothe variable. A timing.- pulse generator I8 triggers a single-sweepsawtootli circuit 9| which produces a highly linear voltage wave `ofincreasing magnitude starting at time to, as represented in the curve AAof Fig. 9. A zero-adjusting circuit 92 is included `to provide for theinitiation of the saw-tooth wave at Yaero voltage at the .time tu. Thiscircuit comprises a .coupling condenser `93 included in series with a.resistor 94 across the output circuit of the 4unit 9|, the resistorhaving coupled in shunt thereto a diode 95 having a grounded anode. Asuitable single-sweep saw-tooth circuit is disclosed in Principles ofRadar, published by McGraw-Hill Fublishing Co., Inc., New York, N. Y.(second edition, 1946) pages 3-2(), Fig. 10.

At a time t1, the saw tooth represented by curve AA .of Fig. V9 reachesa value E1' as determined by a source I6 of `voltage E1. A comparisoncircuit 3 I then develops a potential pulse adapted to cause thegeneration of a potential pulse in a univibrator 15 having a wave formsuitable `.for `energizing another single-sweep saw-tooth circuit 91.VThe latter` then develops a saivetooth voltage starting at the time tiand represented by the curve BB of Fig. 9. Another zero-adjustingcircuit `9B having a similar coupling condenser 99, resistor |00, anddiode |0| provides a zero reference voltage Vfor the saw-tooth wave ofcurve BB. The output circuit of the adjusting circuit 918 is connectedto the switch blade of a singlepole double-throw switch |03, while theoutput circuit of the adjusting circuit 92 is coupled not only to thecomparison circuit 3 I but also tothe switch blade of a single-poledouble-throw switch lII'M operated Vin unicontrol with the switch |03.

When the switches ID3 and |04 are in a position to contact their switchpoints s, the sawtooth circuit 91 is coupled through the adjustingcircuit 98 to a sampling circuit 33, to an output circuit of which isconnected a voltmeter 54. The saw-tooth circuit 9| is coupled throughthe ad- `iusting circuit 92 to a comparison circuit 3| controlled from asource 34 of voltage E2'. When the saw-tooth potential represented "bycurve AA reaches the voltage E2', the comparison circuit 3| produces apotential pulse which is utilized to trigger a pulse generator 32 toproduce a pulse suitable for actuating the sampling circ-uit 33. Thelatter then samples the voltage of the sawtooth circuit 51 to derive thecorresponding voltage E2 which is indicated bythe voltmeter 54.

When the circuits 9| and 91 are so adjusted that the saw-tooth waves ofvoltage produced by them have the same slopes, the rate of increase ofvoltage in each of the saw-tooth circuits is the same, so that tlieincrement of voltage in each of the circuits during the time intervalfrom .ti to t2 is the same. Stated mathematically, this corresponds tothe equation:

E2"=E2'E1 (32) The values of the voltages in Equation 32 may Irepresentthe corresponding parameters of the equation:

so that the Fig. 8 arrangement is useful to perform subtractions.

By moving the switches |03 and |04 to close their switch points a, theFig. 8 arrangement is useful to perform additions. In this case, thevoltage of the saw-tooth circuit 91 is compared in a comparison circuit3| with a voltage from a source 64 of voltage E2 t0 determine the timet2 by reference to the circuit 91 also as illustrated by the curves ofFig. 9. At the time t2, the voltage of the saw-tooth circuit 91 reachesthe voltage E2 and a poential pulse is developed in the comparisoncircuit 3|" to trigger a pulse generator 32" which in turn actuates asampling circuit 33". The latter circuit is coupled to the saw-toothcircuit 9| through the adjusting circuit 92 to derive therefrom avoltage E2', which is the voltage present in the circuit 9| at the timet2. This voltage is indicated on a voltmeter 54".

Since the dependent variable in the circuit arrangement just describedis the voltage E2 of the saw-tooth circuit 9|, the Equation 32 may berearranged in the form:

The values of the voltages in Equation 34 may represent the parametersof the equation:

Substantially linear time functions such as those developed in thesaw-tooth circuits of the Fig. 8 arrangement also may be used formultiplying a number by a factor greater or less than unity. Anarrangement of this type is represented by the block diagram of Fig. 10and the curves of Fig. 1l. A timing-pulse generator I8 causes theinitiation at a time to of a substantially linear saw-tooth wave ofvoltage in a reference saw-tooth circuit 9| provided with azero-adjusting circuit 92.' Simultaneously with the initiation of thissaw-tooth Wave of voltage, another saW- tooth wave of voltage isinitiated by the same timing pulse in a similar saw-tooth circuitY 91provided with a zero-adjusting circuit 98. The wave forms of thevoltages developed in the two circuits 92 and 93 are represented by thecurves CC and DD, respectively, of Fig. 11. The adjusting circuit 92 iscoupled to a comparison circuit 3| controlled by a source 34 of voltageE1. When the voltage of the circuit 9| reaches the voltage E1', at atime which may be designated t1, the comparison circuit 3| triggers apulse generator 32 to'develop a potential pulse suitable for actuating asampling circuit 33 to which the adjusting circuit 98 is coupled. Thus,the voltage E1 then present in the saw-tooth circuit 91 is derived inthe sampling circuit 33 and may be indicated on a voltmeter 54.

The voltage developed in the saw-tooth circuit 9| and its adjustingcircuit 92 may be expressed by the equation: E==k't (36) The values E",

Ykl, W

and E may represent the parameters of the equation:

x=ay (38) An arrangement of this type may be used, for example, tomultiply the parameter 1j by any convenient constant a such as l0. Thusif a computer of the type described is used to solve an equationrepresented by the voltages of Equation l0 hereinabove, and the range ofvoltages over which the voltage E2 in that equation may vary is suchthat the voltage E2 is always quite small, this voltage may bemultiplied by any constant such as 10 by using a computer of the Fig. 10type. The resulting voltage is then applied as a source oi voltage E2 ofthe next higher order of magnitude to the computer for performingmultiplication, as in computation involving an equation of the typerepresented by Equation ll. The product e read from the computer thenmust be divided by the constant a employed, for example 10, to obtainthe `correct solution. The Fig. l0 arrangement has the advantage of afixed accuracy no matter when the time t1 occurs along the saw tooth,and may be made to multiply or divide by large factors because of thelinearity of the circuits available for generating voltages of sawtoothWave form. Since the amplitudes of the saw-tooth wave forni increaserather than decrease with time, the only limit to the voltages which maybe measured is imposed by the duration of the saw-tooth Wave ofvoltage'itself.

In the embodiment of the invention represented by the block diagram ofFig. 12, one of the computation eiects, specifically the effectdeveloped by the reference circuit, varies as a time function analogousto an equation to be solved. A timing-pulse generator i8 causes theenergization of both a single-sweep saw-tooth circuit 91 and, through anenergizing circuit I3, the time-constant circuit R', C of` a unit Bothof the units il and 91 are energized at a time which is indicated to inthe explanatory curves of Fig. 13. Prior to the time to the circuit I!has been charged to a voltage En as determined by the adjustment of asource I2 of such a voltage. The resulting exponentially decayingvoltage in the circuit Il is represented by curve FF of Fig. 13, Whilethe saw-.tooth circuit 9.1 is provided with a zero-adjusting circuit 93which develops a voltage represented by curve GG of Fig. i3. When thevoltage of the circuit reaches a value E1', as determined by a source36, of voltage El', a comparison circuit 3| develops a potential pulseadapted to trigger a pulse generator 32 which in turn provides apotential pulse for actuating a sampling circuit 33 having an inputcircuit coupled to the adjusting circuit 98. if the circuit i reachesthe voltage E1 at a time ti, the voltage E1" in the adjusting circuit 98is derived at that time in the sampling circuit 33 and is indicated onthe voltmeter E.

To illustrate the use of the Fig. l2 arrangement, let it be assumed thatit is required to solve an equation of the form:

The voltage developed in the time-constant circuit I is given by theexpression:

The voltage developed bythe saw-tooth circuit 91 4.EVEN Efmoie'n '0' 43)Equation Li3 maybe transformed into the form:

EL `l efikfRfCI-* `(44) which in turn may take lthe form:

Ell.'

Vf-lOg-Eff 45) If the source '30, Fig, 12,;is adjusted to give a voltage`E1 equal to unity, `Equation 45 becomes:

vEi"=Cf1CygrEf|f `(dm3) where cercano' 47) If the circuit constantsofthe circuits I| and 91 are chosen so that C Aequals' unity, --thevoltage E1" represents the 'natural iogarithm of the voltage En'. If theconstant C rof'itquation i6 :is lmade equal to logt e, then:

where b is any suitable ibase such ias/)the base 10 of the commonIlogarithms By reversing .the positions of `the `time-constantcircuitlil :and the saw-tooth circuit l'911, with respect tothe comparisonIcircuit 3| and sampling circuit 33, the voltage E1" may be made 4torepresent -the .dependent variable .andthe-voltage E1 to represent theindependent variable, so that rEquation 48 takes the form:

Thus the Fig. 12 arrangement may be used for ending iogarithmsandantilocarithms :to any .desired base.

The arrangement :of LFis. ad is useful 'to solve an equation lof the.Iormz or, conversely:

`:cie-tisin-fly (5,1)

In the Fig. 14 arrangement, fthe ltiming-pulse generator I8 is coupledthrough a `single-sweep saw-tooth circuit -91 to a zero-adjustingcircuit 02. The output `circuit -of the latter `coupled to aphase-control circuit `which-includes `a voltage divider ||2 connected-across serially arranged sources of unidirectional potential -shown asbatteries '|10 and LH, `the `:Iuriction of the batteries being coupledtothe-cathode ofthe `diode rectier 95 included Vin vthe zero-adjustingcircuit 92. The movable contact arm of the potential divider |`|2 isconnected to a xed contact T of a singlepole `double-throwswitch .|03and also to a fixed contact AT of a lsingle-pole double-throw switch|05. The switch 2bjl'ade of the switch |03 viscoupled `to thehputcircutof 18 a comparison kcircuit 3|. while the switch blade of the switch |04is coupled to the input circuit of a sampling circuit 33.

The output circuit Aof the timing-pulse generator I8 is also coupled toa sine-Wave ringing generator |05 which includes a triode vacuum tube|06 having `an anode `electrode coupled to a source `of .energizingpotential, indicated as +B, and a cathode electrode coupled to groundthrough a parallel-resonant circuit comprising an inductor |08 and ashunt-connected condenser |01. An inductor ||2 is adjustably coupled tothe inductor |08, and may be selectively coupled through a xed switchcontact T of the switch |04 to the input circuit of the sampling circuit33 or through a fixed switch contact AT of the switch |03 `to the inputcircuit of the comparison circuit 3|.

Considering now the operation of the arrangement j ust described, andreferring to the curves of Fig. 15, the signal of periodic-pulse waveform generated by the generator |8 and represented by curve HH isapplied to the saw-tooth circuit 91 to initiate a signal of saw-toothwave form represented .by curve JJ. Upon translating this signal throughthe zero-,adjusting circuit 92 and the phase-control circuit |09, thereis added to the signal a voltage Eo which is adjustable in magnitude andpolarity with adjustment of the potential divider ||2 so that thevoltage applied to the switch contacts T and AT of the respectiveswitches |03 and |04 may have` an amplitude range as indicated yby thebroken-line curves associated with curve JJ.

The signal o f periodic-pulse wave form generated `by the generator :I8is also applied to the sine-wave ringing `generator |05. The vacuum tube|06 is biased to anode-current cutoii at the trailing edge of eachpulse. The consequent sudden conduction of space current through thetube |06 shocks the inductor |08 and condenser |01 into oscillation atthe frequency:

L=the value of inductance of the inductor |08 C-V-the value ofcapacitance of the condenser |01.

The oscillations thus developed in the resonant circuit |01, |08 arerepresented by curve LL and are coupled into the inductor ||2 with amagnitude dependent upon the value of inductive coupling between theinductors |08 and ||2, and the oscillations induced in the inductor ||2are applied to the switch contacts T and AT of the respective switches|04 and |03.

In solving an equation of the form represented by Equation 50, theswitches |03 and It are moved to close their contacts T so that thephase-.adjusting circuit |09 is coupled to the comparison circuit 3| andthe inductor i|2 is coupled to the sampling circuit 33. When the voltagetranslated through the phase-control circuit |08 equals the voltage E1at time t1, or equals EirL-Eo at the time tii, the triggered pulsegenerator 32 generates a pulseof `short duration as represented by thecurve KK. This generated pulse operates the sampling circuit 33 todeliver to the voltmeter `54 the voltage applied to the input circuit-of the unit 33 from the inductor ||2. Meanwhile, the sine-wavegenerator |05 initiates the generation at time tu of oscillations in itsresonant circuit |01, |08. The voltage thus applied to the sampling'circuit 33. at anytime t is therefore given by the relation:

E2=E3 Sill ZIF However, the time t is determined by the linear saw-toothvoltage appearing-at the terminal T ofthe switch |03 when this voltagewas equal to E1, so that the time t is given by the relation:

Emmi- LEO 54) where K=the slope of the linear saw-tooth voltagegenerated by the unit 9'|-.

The value of the timetl is thus seen from Equation 54 to have the value:

:EFFEO When this value t1 is substituted in Equation 53,

the voltage sampled'by the sampling circuit 33` is given by therelation:

Y E2=sm (Einst) 57) which is of the form:

'J=sin (sq-.0) (58) by letting y=E2; :r=E1; and 0=Eo. The voltage E2thus measured by the voltmeter 54 is represented in Fig. by the curveMM.

Assume now that the switches |03 and |04 are moved to close theircontacts AT. The comparison circuit 3|' now operates at the moment whenthe oscillatory voltage applied thereto from the inductor ||2 is equalto the voltage E1, and the sampling -circuit 33 measures at that timethe value of the saw-tooth 'voltage applied to the latter from thephase-adjusting circuit |09. In this case, we obtain from Equation 53the value of the time t1 when E3' is'equal to Ez as follows:

...1 9 rl Y" Y t1-27rf sin E /E (5779) which when substituted intoEquation 54 becomes: y

E1=- Sin-1 E12/E3 :FEO (60) 21rf Y and by making K /21rf and Ea eachequal to unity, Equation 60 becomesof the form:

VIt will be apparent from the-above description of the invention that anelectrical computer involving the invention is applicable generally tosolving equations the solutions of which involve the common mathematicalrelationships and yet is one free of mechanical moving parts, is compactand of small weight, and is capable of computation at high speeds. Thecomputer of the invention is capable of continuously and rapidlyrecalculating a problem involving parameters subject to change, and iswell adapted to perform algebraic operations in which all of theindependent and dependent variables are represented by voltages referredto a conventional reference datum voltage. r

While there have been described what are at present considered to be thepreferred embodiments of this invention', it will be obvious to thoseskilled'in the 4art`v` that various changes andmodications may be madetherein without departing from the invention, and it is, therefore,aimed to cover all such changes and modifications as fall within thetrue spirit'and scope of the invention.

What is claimed is :Y 1. An electrical computer for solving equationsinvolving known and unknown parameters comprising: an electricalreference circuit having circuit elements With resistance and reactancevalues so proportioned as todevelop an electrical effect varying as apredetermined time function the value of which at some time represents aknown parameter of an equation to be solved; a source of potentialhaving a value representative of a parameter of said equation; means forutilizing said potential of said source and said effect at saidpredetermined time to develop a control effect; an electrical controlledcircuit having circuit elements with resistance and reactance Values soproportioned as to developanother electrical efect having a valueelectrically independent of the value of said first-mentioned effect andVarying as a predetermined time function a value of which at some timeis related to the value of an unknown parameter of said equation; meansresponsive to said control effect for evaluating said other effect toderive a resultant eifect which represents an unknown parameter of saidequation; and means forutilizing said resultant effect. 2. An electricalcomputer for solving equations involving known and unknown parameterscomprising: an electrical reference circuit having circuit elements withresistance and reactance values so proportioned as to develop anelectrical effect varying as a predetermined time function which at aninstant'of time determined by an independent variable known parameter ofan equation to be solved reaches a value representing said knownparameter; a source of potential having a value representative of saidindependent variable known parameter; means for utilizing said potentialof said source and said eifect at said predetermined time to develop acontrol effect; an electrical controlled circuit having circuit elementswith resistance and reactance values so proportioned as to developanother electrical effect having a value electrically independent of thevalue of said first-mentioned effect and Varying as a predetermined timefunction a value of which at some time is related to the value of anunknown parameter of said equation; means responsive to saidcontrolelfect for evaluating said other effect to derive a resultanteffect which represents an unknown parameter of said equation; and meansfor utilizing said resultant effect. 3. An electrical computer forsolving equations involving known and unknown parameters comprisingzanelectrical reference circuit having circuit elements with resistance andreactance Values so proportioned as to develop an electrical effectvarying as apredetermined time function the value of which at some time`represents a known parameter of an equation to be solved; a source ofpotential having @Value representative of a parameter of said equa-tion;means for utilizing said potentialof sadsource and ys aid eifect at saidpredetermined timetodevelop a control effect; an electrical controlledcircuit having circuit elements with resistancel and reactance values soproportioned as tode'velop another electrical effect having a valueelectrically independent of the value of said first-mentioned effect andvarying as a predetermined time function a value of which at sometime isrelated. tothe Value of an unsesam known parameter of said equation, vatleast one of said effects varying as atime function analogous to theequation to be solved; means responsive to said control effect forevaluating said other effect to derive a resultant effect whichrepresents an unknown parameter of said equation; and means forutilizing said resultant-effect.

4. An electrical computer forsolving equations involving known and`unknown parameters comprising: a first impedance network having circuitelements with resistance and reactance values so proportioned as todevelop an electrical effect varying as a predetermined time functionthe value of which at some time represents a known parameter of anequation to be solved; a source of potential having a valuerepresentative of a parameter of said equation; means for utilizing saidpotential of said source and-said effect at said predetermined time todevelop a control effect; a second impedance network having'circuitelements with resistance and reactance values so proportioned as todevelop another electrical effect having a value electricallyindependent of the value of said first-mentioned effect and varying as apredetermined time function a value of which at some time is related tothe value of an unknown parameter of said equation, at least one of saidfunctions varying as a substantially Vlinear function of time; meansresponsive to said control effect for evaluating said other effect toderive a resultant effect which represents an unknown parameter of saidequation; and means for utilizing said resultant effect.

5. An electrical computer for solving equations involving known andunknown'parameters comprising: an electrical reference circuit havingcircuit elements with resistance and reactance values so proportioned asto develop an electrical effect varying as a predetermined substantiallylinear time function the value of which at some time represents a knownparameter of an equation to be solved; a source of potential having avalue representative of a parameter of said equation; asignal-comparison circuit for combining said potential of said sourceand said effect at said predetermined time to develop a control effect;an electrical controlled circuit having circuit elements with resistanceand reactance values so proportioned as to develop another electricaleffect having a value electrically independent of the value of saidfirst-mentioned effect and varying as a predetermined substantiallylinear time function a value of which at some 'time is related to thevalue of an unknown parameter of said equation; means responsive to saidcontrol effect for evaluating said other effect to derive a resultanteffect which represents an unknown parameter of said equation; and meansfor utilizing said resultant effect. A

46. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference circuit having.circuit elements with resistance and reactance values so proportionedas to develop an electrical effect varying as a predetermined timefunction the value of which at some time represents a known parameter ofan equation to be solved; a source of potential :having a valuerepresents/tive of a parameter of said equation; means coupled to saidreference circuit and said source for utilizing said potential of saidsource and said effect at said predetermined time to develop a `controleffect; an electrical controlled circuit having circuit elements withresistance and reactance values so proportioned as to develop anotherelectrical effect having a value electrically independent of the valueof said firstmentioned effect and varying `as a predetermined timefunction a value of which at some time is related to the value of anunknownparameter of said equation, at least one of said effects varyingas a trigonometric function of time; means responsive to said controleffect for evaluating said other effect to derive a resultant effectwhich represents an unknown parameter of said equation; and means forutilizing said resultant effect.

7. An electrical computer for solving equations involving known andunknown parameters comprising: a first time-constant circuit havingcircuit elements with resistance and reactance values so proportionedas'to develop an electrical effect varying as a predetermined timefunction the value of which at sometime represents a known parameter ofan equation to be solved; `a source of potential having a valuerepresentative of a parameter of said equation; an electron-dischargedevice for comparing said potential of said source and said effect atsaid predetermined timeto develop a control effect; a secondtimeconstant circuit having circuit elements with resistance andreactance values so proportioned as to develop another electrical effecthaving a value electrically independent of the value of saidfirst-mentioned effect and varying as a predetermined time function avalue of which at some time is related to the value ofanunknownparameter of said equation, at least one of said effectsvarying as an exponential function of time; means responsive to saidcontrol effect for evaluating said other eifect to derive a resultanteffect which represents an unknown parameter of said equation; and meansVfor utilizing said resultant effect.

8. An electrical computer for solving equations involving known andunknown parameters comprising: a first'adjustable time-constant circuithaving circuit elements with resistance and reactance values soproportioned as to develop an electrical effect varying as apredetermined time function the value of which at some time represents aknown parameter of an equation to be solved; a source of potentialhaving a value representative of a parameter of said equation; means forutilizing said `potential of said source and said effect at saidpredetermined time to develop a control effect; a second adjustabletimeconstant circuit having circuit elements with resistance andreactance `values so proportioned as to develop another electricaleffect having a value electrically independent of the value of saidfirstmentioned effect and varying as a predetermined time function avalue of which at some time is related to the value of an unknownparameter of said equation, at least one of said effects varying as anexponentially decaying .function of time; means responsive to saidcontrol effect for evaluating said other effect to derive a resultanteffect which represents an unknown parameter of said equation; and meansfor utilizing said resultant effect.

9. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference circuit havingcircuit elements with resistance and reactance values so proportioned asto develop an electrical effect varying as a predetermined exponentialtime function the value of which at some time represents a knownparameter of an equation to be solved; a source of potential having avalue representative of `a parameter of :said equation;

means for utilizing said potential'of said source and said effect atsaid predetermined time to develop a control effect; an electricalcontrolled circuit having circuit elements with resistance and reactancevalues so proportioned as to develop another electrical effect having avalue electrically independent of the value of said firstmentionedeffect and varying as a predetermined exponential time function a valueof which at some time is related to the value of an unknown parameter ofsaid equation; means responsive to said control effect for evaluatingsaid other effect to derive a resultant effect which represents anunknown parameter of said equation; and means for utilizing saidresultant effect.

10. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference circuit havingcircuit elements with resistance and reactancevalues so proportioned asto develop an electrical effect varying as a predetermined exponentialdecaying time function the value of which at some time represents aknown parameter of an equation to be solved; a source of potentialhaving a value representative of a parameter of said equation; means forutilizing said potential of said source and said effect at saidpredetermined time to develop a control effect; an electrical controlledcircuit having circuit elements with resistance and reactance values soproportioned as to develop another electrical effect having a valueelectrically independent of the value of said firstmentioned effect andvarying as a predetermined exponential decaying time function a value ofwhich at some time is related to the value of an unknown parameter ofsaid equation; means responsive to said control effect for evaluatingsaid other effect to derive a resultant effect which represents anunknown parameter of said equation; and means for utilizing saidresultant effect.

1l. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference circuit havingcircuit elements with resistance and reactance values so proportioned asto develop an electrical effect varying as a predetermined time functionthe value of which at some time represents a known parameter of anequation to be solved; a source of potential having a valuerepresentative of a parameter of said equation; means for utilizing saidpotential of said source and said effect at said predetermined time todevelop a control effect; an electrical controlled circuit havingcircuit elements with resistance and reactance values so proportioned asto develop another electrical effect having a value electricallyindependent of the value of said firstmentioned effect and varying as apredetermined time function a value of which at some time is related tothe value of an unknown parameter of said equation, at least one of saidcircuits being an energy-storage network; means responsive to saidcontrol effect for evaluating said other effect to derive a resultanteffect which represents anunknown parameter of said equation; and meansfor utilizing said resultant effect.

l2. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference energy-storagenetwork having circuit elements with resistance and reactance values soproportioned as to develop an electrical effect varying as apredetermined exponentially decaying time function the value of which atsome time represents a known parameter of an equation to be solved; a

source of potential having a'value representative of a parameter of saidequation; means for utilizing said potential of said source and saideffect at said predetermined time to develop a control effect; anelectrical controlled energy-storage network having circuit elementswith resistance and reactance values so proportioned as to developanother electrical effect having a value electrically independent of thevalue of said firstmentioned effect and varying as a predeterminedexponentially decaying time function a value of which at some time isrelated to the value of an unknown parameter of said equation; meansresponsive to said control effect for evaluating said other effect toderive a resultant effect which represents an unknown parameter of saidequation; and means for utilizing said resultant effect.

13. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference circuit havingcircuit elements with resistance and reactance values so proportioned asto develop upon energization an electrical effect varying as apredetermined time function the value of which at some time after saidenergization represents a known parameter of an equation to be solved; asource of potential having a value representative of a parameter of saidequation; means for utilizing said potential of said source and saideffect at said predetermined time to develop a control effect; anelectrical controlled circuit having circuit elements with resistanceand reactance values so proportioned as to develop upon energizationthereof at a time related to the time of said energization of saidreference circuit another electrical effect having a value electricallyindependent of the value of said rst-mentioned effect and Varying as apredetermined time function a value of which at some time is related tothe value of an unknown parameter of said equation;` means responsive tosaid control effect for evaluating said other effect to derive aresultant effect which represents an unknown parameter of said equation;and means for utilizing said resultant effect.

14. An electrical computer for solving equations involving known andunknown parameters comprising: an electrical reference circuit havingcircuit elements with resistance and reactance values so proportioned asto develop upon energization an electrical effect varying as apredetermined time function the value of which at some time after saidenergizaticn represents a known parameter of an equation to be solved;means for repeatedly energizing said reference circuit; a source ofpotential having a value representative of a parameter of said equation;means for utilizing said potential of said source and said effect atsaid predetermined time after each of said repeated energizations todevelop a'control effect; an electrical controlled circuit havingcircuit elements with resistance and reactance values so proportioned asto develop upon each similarly repeated energization thereof anotherelectrical effeot having a value electrically independent of the valueof said rst-mentioned effect and varying as a predetermined timefunction a value of which at some time is related to the value of anunknown parameter of said equation; means responsive to each of saidcontrol effects for evaluating said other effect to derive a resultanteffect which represents an unknown parameter of said equation; and meansfor utilizing said resultant effect.

, 15,;Anv electrical computer for solving equations involving knownand'unknown parameters comprising: an electricalreference circuit havingcircuit elements with resistance and reactance values so proportioned asto develop upon energization an electrical effect varying asapredetermined time function which, at an instant of time after saidenergization asfcletermined by an independent variable known parameterof an equation to be solved, reaches a value represent ing said knownparameter; means for repeatedly energizing said reference circuit; asource of potential having a value representative of a parameter of saidequation; means for utilizing said potential of said source and saideffect at said predetermined time after each of said repeatedenergizations to develop a control effect; an electrical controlledcircuit having circuit elements with resistance and reactance values soproportioned as to develop upon each similarly repeated energizationthereof another electrical effect having a value electricallyindependent of the value of said first-mentioned effect and varying as apredetermined time function a value of which at some time is related tothe value of an unknown parameter of said equation variable withvariations of said independent variable known parameter; meansresponsive to each of said control effects for evaluating said othereffect to derive a resultant effect which represents an unknownparameter of said equation; and means for utilizing said resultanteffect.

16. An electrical computer for solving equations involving known andunknown parameters comprising: a plurality of electrical circuits eachwith resistance and reactance values so proportioned as to develop uponenergization a measurable electrical effect having a Value which is apredetermined function of time, said known and unknown parameters beingrepresented by values assumed by the time functions of said effects atrelated times; means for energizing at least one of said circuits toproduce at least one of said effects; a source of potential having avalue representative of a known parameter of said eduation fordetermining a first instant of time at which the value of the effectproduced by said one circuit represents one of said known parameters;means for utilizing said potential of said source and one of saideffects at said first instant of time to develop a control effect; saidenergizing means also being so proportioned as to initiate energizationof at least one other of said circuits at a time related to said firstinstant of time to produce at least another of said electrical effectshaving a value independent of the value of said one of said effects;means ef fectively responsive to said control effect for determining asecond instant of time at which the value of another of said eectsrepresents an unknown parameter; and means for utilizing said othereffect representative of said unknown parameter.

17. An electrical computer for solving equations involving known andunknown parameters comprising: a plurality of electrical circuits eachwith resistance and reactance values so proportioned as to develop uponenergization a measurable electrical effect having a value which is apredetermined function of time, at least one of said effects having avalue varying as a substantially linear function of time and said knownand unknown parameters being represented by values assumed by the timefunctions of said effects at related times; means for energizing atleast one of said circuits to produce at least .one

of said effects; a source of potential having' a value representative ofa known parameter of said equationfor determining a first instant oftime at which-the value of the effect produced by said one circuitrepresents one of said known parameters; means for utilizingsaidpotential of said source and one of said effects at said rst instantof time to develop a control effect; said energizing means also being soproportioned as to initiate energization of at least one other of saidcircuits at a time related tosaid first instant of time to produce atleast another of said electrical effects having a value independent ofthe value of said one of said effects; means effectively responsive tosaidV control effect fordetermining a second instant of time at whichthe value of another of said effects represents an unknown parameter;and means for utilizing said other effect representative of said unknownparameter.

18. An electrical computer for solving equations involving known andunknown parameters comprising: a plurality of electrical circuits eachwith resistance and reactance values so proportioned as to develop uponenergization of measurable electrical effect having a value which is apredetermined function of time, at least one of said effects having avalue varying as a trigonometric function of time and said known andunknown parameters being represented by values assumed by the timefunctions of said effects at related times; means for energizing atleast one of said circuits to produce at least one of said effects; asource of potential having a value representative of a known parameterof said equation for determining a rst instant of time at which thevalue of the effect produced by said one circuit represents one of saidknown parameters; means for utilizing said potential of said source andone of said effects at said first instant of time to develop a controleffect; said energizing means also being so proportioned as to initiateenergization of at least one other of said circuits at a time related-to said rst instant of time to produce at least another of saidelectrical effects having a value independent of the value of said oneof said effects; means effectively responsive to said control effect fordetermining a second instant of time at which the value of another ofsaid effects represents an lunknown parameter; and means for utilizingsaid other effect representative of said unknown parameter.

19. An electrical computer for solving equations involving known andunknown parameters comprising: a plurality of electrical circuits eachwith resistance and reactance values so proportioned as to develop uponenergization a measurable electrical effect having a value which is apredetermined function of time, at least one of said effects having avalue varying as an exponential function of time and said known andunknown parameters being represented by values assumed by the timefunctions of said effects at related times; means for energizing atleast one of said circuits to produce at least one of said effects; asource of potential having a value representative of a known parameterof said equation for determining a rst instant of time at which thevalue of the effect produced by said one circuit represents one of saidknown parameters; means for utilizing said potential of said source andone of said effects at said first instant of time to develop a controleffect; said energizing means also being so proportioned as

